Respecting the matter in general relativity

General relativity is a theory which describes space, time and the gravitational field in terms of a Lorentzian metric . A complete understanding of the gravitational field requires an understanding of the matter sources which generate it. In the Einstein equations, , the left hand side depends only on and is a feature of the geometry alone. On the other hand the right hand side, the energy-momentum tensor, depends not only on the metric but also on some matter fields. The right hand side of Einstein’s equations seems to have suffered from bad press coverage from an early stage. Einstein himself is often quoted as having said that the left hand side of his equations is made of marble while the right hand side is made of wood. I do not have a source for this quote – if anyone reading this does I would be grateful to hear about it. In this post I want to suggest treating that humble right hand side with more respect. If I lived in a palace made of marble with beautiful wooden furniture then I might be more impressed by the marble than by the wood. I would nevertheless do my best to prevent little boys from carving their initials into the furniture with penknives or the cat (much as I love cats) from using it as an accessory for the care of its claws.

The left hand side of the Einstein equations is universal within general relativity – it is always the same, no matter which type of physical situation is to be described. On the other hand the nature of the matter fields depends very much on what physical situation is to be described and what aspects of it are to be included in the description. It is necessary to make a choice of matter model. What is remarkable is that there is a large variety of choices which, in conjunction with the Einstein equations, lead to a consistent closed system of equations which bears no traces of the fact that other physical effects have been omitted. In fact there are three related choices which have to be made to set up the mathematical model in any given case. The first is the matter fields themselves – what kind of geometrical objects are they? The second is the expression which defines the energy-momentum tensor in terms of the matter fields and the metric. The third is the system of equations of motion which describe the dynamics of the matter. Note that in general the energy-momentum tensor depends explicitly on the metric. It is not possible to define an energy-momentum tensor unless the spacetime geometry is given. The same is true in the case of the equations of motion of the matter. They also contain the metric explicitly. Without the metric even the nature of the matter fields themselves can become ambiguous. Which positions should we choose for the indices of a tensor occurring in the description of the matter fields? From a physical point of view it is clear why the metric is necessary in so many ways. The mathematical model must be given a physical interpretation which involves the consideration of measurements. In the absence of a given geometry there is no way to talk about measurements.

When a matter model has been chosen the basic equations which are to be solved are the Einstein-matter equations, i.e the Einstein equations coupled to the equations of motion for the chosen type of matter. The unknowns are the metric and the basic matter fields. For any reasonable choice of matter fields the energy-momentum tensor has zero divergence as a consequence of the equations of motion. However the equations of motion in general contain more information than the divergence-free property of the energy-momentum tensor. For more discussion of these things together with examples see Chapter 3 of my book. I emphasize that solving the equations describing the physical situation within the given model means solving both the Einstein equations and the equations of motion of the matter. This is too often neglected in the literature. A particular danger occurs when the solutions under consideration are of low regularity. If the Einstein equations do not make sense pointwise then it should be checked that they hold in the sense of distributions. For solutions which lack regularity on a hypersurface this is expressed in the junction conditions and it is common in the literature to check that they hold. The equations of motion should also be satisfied in the sense of distributions and this is often ignored. When I use the phrase ‘in the sense of distributions’ here this is just a shorthand since the equations are nonlinear. The correct statement is that it is necessary to think carefully about the sense in which the equations are satisfied.

An example may help to make the importance of the issue clear. At the GR12 conference in Boulder in 1989 there was a heated discussion of the question, whether colliding plane waves can give rise to spontaneous creation of matter. (I emphasize that this discussion was in a purely classical context. Quantum theory was not being taken into account.) This kind of creation of matter sounds ridiculous from a physical point of view. Nevertheless people exhibited ‘solutions’ which showed this type of effect. Their mistake was that they had only verified those things which I said above were usual in the literature. They had not considered whether the equations of motion of matter were satisfied. If the equations of motion are ignored, it is not surprising that arbitrary things can happen.

Uwe Brauer and Roger Bieli have now provided me with some information about the quote concerning marble and wood. A comment of this type was published by Einstein in the Journal of the Franklin Institute in 1936. I reproduce the original German text and the contemporary translation:

Sie gleicht aber einem Gebäude, dessen einer Flügel aus vorzüglichem Marmor (linke Seite der Gleichung), dessen anderer Flügel aus minderwertigem Holze gebaut ist (rechte Seite der Gleichung). [“But, it is similar to a building, one wing of which is made of fine marble (left part of the equation), but the other wing of which is built of low grade wood (right side of equation).”]

Great post on Matter in GR. I really enjoy your blog, especially because you post on PDEs in general, not just GR. Anyway, my question is, what is the status of giving a mathematical definition of mass? I know there are various definitions of mass: ADM, quasi local, etc.

Also, we all know GR is a fantastically successful physical theory but also incomplete in some aspects. A classic example is the expansion of the universe and so called “dark matter, dark energy.” But what about the fact that GR cannot incorporate intrinsic spin. What efforts have been made on this front?

The most important concept of mass or energy in general relativity is that of ADM and it is global. The general difficulty is that this total energy cannot be expressed in a useful way as the integral of an energy density. The best that can be done is to associate some kind of quasilocal mass to certain regions. Various definitions have been proposed and, while I do not have a good overview of the subject, it seems to me that there is no clear leader. I think that the most interesting mathematical developments in this area are those related to the Penrose inequality. A question which should be asked is: why should we be interested in trying to define a concept of quasilocal mass? My (no doubt biased) answer is: to try to find new PDE estimates for the Einstein evolution equations.

My preferred solution to the dark matter problem would for someone to detect particles in terrestrial experiments which could constitute dark matter. There is a claim of observations of this type by the DAMA collaboration at the Gran Sasso laboratory but apparently the results could not be confirmed by other experiments up to now. If suitable observations of this type could be made then dark matter would not be a problem for general relativity. Concerning dark energy, the data on accelerated cosmological expansion can be explained by a positive cosmological constant or by some other (more or less exotic) matter field. The huge variety of matter fields which have been suggested in this context are a nuisance for physics – with so many alternative models how do we get a definite prediction – but a nice playground for mathematicians. In any case, finding a model which explains the data is not a problem for general relativity – only uniqueness is difficult. Some people have suggested that accelerated cosmological expansion could be explained without the need for dark energy. It is supposed to arise as a consequence of averaging the nonlinear Einstein equations. My first serious encounter with this subject was in a talk I heard at AEI in 2005. (It was part of the conference ‘Geometry and Physics after 100 years of Einstein’s relativity.’) One of the people who initiated the present interest in this topic was Edward Kolb. At the time of that talk it seemed clear that the original proposal was seriously flawed. This was asserted convincingly by Norbert Straumann during the questions after the talk. Experience shows that when a proposal of this type is shown to be untenable its response is not to die but to mutate. The idea of getting cosmological expansion without dark energy is typical in this respect. Two of the most visible supporters of the idea are Thomas Buchert and David Wiltshire. I missed a talk which Buchert gave at AEI but I got a report on it by Jürgen Ehlers, which was probably more valuable than my own reaction would have been. In fact Jürgen had been talking about the importance of the issue of averaging in cosmology for more than twenty years, while almost nobody else cared. Jürgen was not at all convinced by what Buchert had to say. More recently I heard a talk by Wiltshire which I also did not find convincing. On a conceptual level these explanations seem to be very murky. My guess is that if I just wait then most of the interest in this idea will go away, even if a low level of endemic infection remains.

Well, that is probably a misinterpretation of Jürgen. He was in fact adding a talk
on the subject thereafter, where he supported the importance of curvature
(the source for importance of the backreaction effect), even
for small metric perturbations. His remarks are now published in the paper by
myself, George Ellis and Henk van Elst in the Memorial Issue of GRG.